Sunday, 29 September 2013

Assistant Professor Position at LMU Munich (MCMP)

The  Chair of Philosophy of Science at the Faculty of Philosophy, Philosophy of Science and the Study of Religion and the Munich Center for Mathematical Philosophy (MCMP, http://www.lmu.de/mcmp) at LMU Munich seek applications for an Assistant Professorship with a specialization in (at least) one of the following areas: Philosophy of Psychology, Philosophy of Social Science, Philosophy of Economics, and Philosophy of Neuroscience. The position is for three years with the possibility of extension for another three years. Note that there is no tenure-track option. The appointment will be made within the German A13 salary scheme (under the assumption that the civil service requirements are met), which means that one has the rights and perks of a civil servant. The starting date is October 1, 2014. A later starting date is also possible.

The appointee will be expected (i) to do philosophical research and to lead a research group in her or his field, (ii) to teach five hours a week in at least one of the above-mentioned fields and/or a related field, and (iii) to take on some management tasks. The successful candidate will have a PhD in philosophy and some teaching experience.

Applications (including a cover letter that addresses, amongst others, one's academic background and research interests, a CV, a list of publications, a list of taught courses, a sample of written work of no more than 5000 words, and a description of a planned research project of 1000-1500 words) should be sent by email (ideally everything requested in one PDF document) to office.hartmann@lrz.uni-muenchen.de by November 20, 2013. Hard copy applications are not possible. Additionally, two confidential letters of reference addressing the applicant's qualifications for academic research should be sent to the same address from the referees directly.

For further information, please contact Professor Stephan Hartmann (S.Hartmann@lmu.de).

Thursday, 26 September 2013

The Mathematics of Dutch Book Arguments

Dutch Book arguments purport to establish norms that govern credences (that is, numerically precise degrees of belief).  For instance, the original Dutch Book argument due to Ramsey and de Finetti aims to establish Probabilism, the norm that says that an agent's credences ought to obey the axioms of mathematical probability.  And David Lewis' diachronic Dutch Book argument aims to establish Conditionalization, the norm that says that an agent ought to plan to update in the light of new evidence by conditioning on it.  As we will see in this post, there is also a Dutch Book argument for the Principal Principle as well, the norm that says that an agent ought to defer to the chances when she sets her credences.  We'll look at each of these arguments below.

Each argument consists of three premises.  The second is always a mathematical theorem (sometimes known as the conjunction of the Dutch Book Theorem and the Converse Dutch Book Theorem).  My aim in this post is to present a particularly powerful way of thinking about the mathematics of these theorems.  It is due to de Finetti.  It is appealing for a number of reasons:  it is geometrical, so we can illustrate the theorems visually; it is uniform across the three different Dutch Book arguments we will consider here; and it establishes both Dutch Book Theorem and Converse Dutch Book Theorem on the basis of the same piece of mathematics.

I won't assume much mathematics in this post.  A passing acquaintance with vectors in Euclidean space might help, but it certainly isn't a prerequisite.

The form of a Dutch Book argument


The three premises of a Dutch Book argument for a particular norm $N$ are as follows:

(1) An account of the sorts of decisions a given set of credences will (or should) lead an agent to make.

(2) A mathematical theorem showing two things:  (i) relative to (1), credences that violate norm $N$ will lead an agent to make decisions with property $C$; (ii) relative to (1), credences that satisfy norm $N$ in question will not lead an agent to make decisions with this property $C$.

(3) A norm of practical rationality that says that, if an agent can avoid making decisions with property $C$, she is irrational if she does make such a decision.

In this post, I'll present Dutch Book arguments of this form for Probabilism, Conditionalization, and the Principal Principle.  But I'll be focussing on premise (2) in each case.  There's plenty to say about premises (1) and (3), of course.  But that's for another time.

The Dutch Book argument for Probabilism


The first premise in each Dutch Book argument is the same.  It has two parts:  the first tells us, for any proposition in which the agent has a credence, the fair price she ought to pay for a bet on that proposition; the second tells us the price she ought to pay for a book of bets on a number of different propositions given the price she's prepared to pay for each individual bet.  Thus, we have

(1a)  If an agent has credence $p$ in proposition $X$, she ought to pay $pS$ for a bet that pays out $S$ if $X$ is true and $0$ if $X$ is false.  (In such a bet, $S$ is called the stake.)

(1b) If an agent ought to pay $X$ for Bet 1 and $Y$ for Bet 2, she ought to pay $X+Y$ for a book consisting of Bet 1 and Bet 2.  (This is sometimes called the Package Principle.)

Putting these together, we get the following:  Suppose $\mathcal{F} = \{X_1, \ldots, X_n\}$ is a set of propositions.  And suppose we represent our agent's credences in these $n$ propositions by a vector \[ c = (c_1, \ldots, c_n) \] where $c_i$ is her credence in $X_i$.  And suppose we consider a book of bets $S$ in which the stake on $X_i$ is $S_i$.  Then we can represent this book by the vector \[ S = (S_1, \ldots, S_n) \] Then the price that the agent ought to pay for this book of bets is \[ \sum^n_{i=1} S_ic_i :=  (S_1, \ldots, S_n) \cdot (c_1, \ldots, c_n) = S\cdot c \] where $S\cdot c$ is the dot product of $c$ and $S$ considered as vectors.

Happily, there is also a nice way to represent the payoff of a book of bets $S$ at a given possible world $w$.  Represent that possible world $w$ by the following vector: \[ w = (w_1, \ldots, w_n) \] where $w_i = 1$ if $X_i$ is true at $w$ and $w_i = 0$ if $X_i$ is false at $w$.  Then the payoff of $S$ at $w$ is \[\sum^n_{i=1} S_iw_i := (S_1, \ldots, S_n) \cdot (w_1, \ldots, w_n) : S\cdot w \]  As we will see, these vector representations will prove very useful below.

In this section, we're looking at the Dutch Book argument for Probabilism.

Probabilism  It ought to be that a set of credences $c$ obeys the axioms of mathematical probability.

Let us turn to premise (3) of this argument.  It says that it is irrational for an agent to have credences that lead her to make decisions that will lose her money in every world that she considers possible.  Now, a book of bets loses an agent money if \[\mbox{Payoff} < \mbox{Price}\] But recall from above:  the payoff of a book of bets $S$ at a world $w$ is $S \cdot w$; and the price of that book is $S \cdot c$.  Thus, the agent is irrational if there is a book $S$ such that \[S \cdot w < S \cdot c\] for all worlds $w$.  Equivalently, $S \cdot (w-c) < 0$ for all $w$.

So the Dutch Book Theorem (that is, premise (2)) can be stated as follows:

Theorem 1
(i) If $c$ violates Probabilism, then there is a book $S$ such that $S \cdot w < S \cdot c$ for all worlds $w$ (equivalently, $S \cdot (w-c) < 0$ for all $w$).
(ii) If $c$ satisfies Probabilism, then there is no book $S$ such that $S \cdot w \leq S \cdot c$ (equivalently, $S\cdot (w-c) \leq 0$) for all worlds $w$ and $S \cdot w < S \cdot c$ (equivalently, $S \cdot (w-c) < 0$) for some world $w$.

We now turn to the proof of this theorem.  It is based on two pieces of mathematics:  the first involves some basic geometrical facts about the dot product; the second involves a neat geometric characterization of the credences that satisfy Probabilism.

First, a well known fact about the dot product.  If $u$ and $v$ are vectors in $\mathbb{R}^n$, we have \[ u \cdot v = ||u||\, ||v|| cos \theta\] where $\theta$ is the angle between $u$ and $v$.  Since $||u||\, ||v|| \geq 0$, we have \[u\cdot v < 0 \Leftrightarrow cos \theta < 0\]  And, by basic trigonometry, we have \[u \cdot v < 0 \Leftrightarrow \frac{\pi}{2} < \theta < \frac{3\pi}{2}\]  Thus:
  • To prove Theorem 1(i), it suffices to show that, if $c$ violates Probabilism, we can find a vector $S$ such that the angle between $S$ and $w-c$ is oblique for all worlds $w$.
  • To prove Theorem 1(ii), it suffices to show that, if $c$ satisfies Probabilism, there is no vector $S$ such that the angle between $S$ and $w-c$ is oblique or right for all $w$ and oblique for some $w$.
To do this, we need a geometric characterization of the credences that satisfy Probabilism.  Fortunately, we have that in the following lemma due to de Finetti:

Lemma 1 $c$ satisfies Probabilism iff $c \in \{w : w \mbox{ is a possible world}\}^+$.

where, if $\mathcal{X}$ is a set of vectors in $\mathbb{R}^n$, $\mathcal{X}^+$ is the convex hull of $\mathcal{X}$:  that is, $\mathcal{X}^+$ is the smallest convex set that includes $\mathcal{X}$; if $\mathcal{X}$ is finite, then $\mathcal{X}^+$ is the set of linear combinations of elements of $\mathcal{X}$.

Thus, Lemma 1 says that the vectors that represent the probabilistic sets of credences are precisely those that belong to the convex hull of the vectors that represent the possible worlds.

How does this help? Let's take the case in which $c$ violates Probabilism.  That is, $c$ lies outside the convex hull of the vectors representing the different possible worlds.  Then it is easy to see from Figure 1 below that there is a vector $c^*$ that lies inside that convex hull such that, for a given world $w$, the angle $\theta$ between the vector $c-c^*$ and the vector $w-c$ is oblique.  Thus, if we let $S = c - c^*$, we have Theorem 1(i).

Figure 1: The oval represents the convex hull of the set of vectors that represent the different possible worlds.  If $c$ violates Probabilism, then it lies outside this.  But, by a Hyperplane Separating Theorem, there is a point $c^*$ in the convex hull such that the angle between $c-c^*$ and $x-c$ is oblique for any $x$ inside the convex hull.  Thus, in particular, it is oblique when $x$ is a vector representing a possible world, as required.

Now let's take the case in which $c$ satisfies Probabilism.  That is, $c$ lies inside the convex hull of the vectors representing the different possible worlds.  Then it is easy to see from Figure 2 below that, if $S$ is a vector, then while there may be some worlds $w$ such that the angle $\theta$ between $S$ and $w-c$ is oblique, there must also be some worlds $w'$ such that the angle $\theta'$ between $S$ and $w'-c$ is acute.  Alternatively, it is possible that the angles $\theta$ between $S$ and $w-c$ for all worlds $w$ are all right.

Figure 2: Again, the oval represents the convex hull of the possible worlds.  If $c$ satisfies Probabilism, then it lies inside.
This completes the geometrical proof of Theorem 1, which combines the Dutch Book Theorem and the Converse Dutch Book Theorem.

The Dutch Book Argument for the Principal Principle


The Principal Principle says, roughly, that an agent ought to defer to the chances when she sets her credences.  One natural formulation of this (explicitly proposed by Jenann Ismael and entailed by a slightly stronger formulation proposed by David Lewis) is this:

Principal Principle  It ought to be the case that $c$ is in $\{ch : ch \mbox{ is a possible chance function}\}^+$.

That is, the Principal Principle says that one's credence function ought to be a linear combination of the possible chance functions.

Now, adapting the proof of Theorem 1 above, replacing the possible worlds $w$ by possible chance functions $ch$ (represented as vectors in the natural way), we easily prove the following:

Theorem 2
(i) If $c$ violates the Principal Principle, then there is a book $S$ such that $S \cdot ch < S \cdot c$ for all possible chance functions $ch$.
(ii) If $c$ satisfies Probabilism, then there is no book $S$ such that $S \cdot ch \leq S \cdot c$ for all possible chance functions $ch$ and $S \cdot ch < S \cdot c$ for some possible chance function $ch$.

But what does this tell us?  Well, as before, $S \cdot c$ is the price our agent would pay for the book $S$.  But this time, the other side of the inequality is $S\cdot ch$.  And this, it turns out, is the objective expected payout of $S$, rather than the actual payout of $S$.  Thus, violating the Principal Principle does not necessarily make an agent vulnerable to a true Dutch Book.  But it does lead them to pay a price for a book of bets that is higher than the objective expected value of that book, according to all of the possible chance functions.  And this, we might think, is irrational.  For one thing, such an agent will, with objective chance 1, lose money in the long run.  Thus, in the Dutch Book argument for the Principal Principle, premise (1) is as before, premise (2) is Theorem 2, but premise (3) becomes the following:  It is irrational for an agent to have credences that lead her to pay more than the objective expected value for a book of bets.

The Dutch Book Argument for Conditionalization


Conditionalization is the following norm:

Conditionalization  Suppose our agent has credence $c$ at $t$; and suppose she knows that, by $t'$, she will have received evidence from the partition $E_1, \ldots, E_m$.  And suppose she plans to update as follows:  If $E_i$, then $c_i$.  Then it ought to be that $c_i(-) = c(-|E_i)$ for $i = 1, \ldots, m$.

In fact, the Dutch Book argument for Conditionalization that we will present is primarily a Dutch Book argument for van Fraassen's Reflection Principle, which is equivalent to Conditionalization.  The Reflection Principle says the following:

Reflection Principle  Suppose our agent has credence $c$ at $t$; and suppose she knows that, by $t'$, she will have received evidence from the partition $E_1, \ldots, E_m$.  And suppose she plans to update as follows:  If $E_i$, then $c_i$.  Then it ought to be that:
(i) $c_i(E_i) = 1$ for $i = 1, \ldots, m$;
(ii) $c$ is in $\{c_i : i = 1, \ldots, m\}^+$.

 That is, Reflection says that an agent's current credences ought to be a mixture of her planned future credences.  Since Reflection and Conditionalization are equivalent, it suffices to establish Reflection.

Here is the theorem that provides the second premise of the Dutch Book argument for Reflection:

Theorem 3
(i) Suppose $c, c_1, \ldots, c_n$ violate Reflection.  Then there are books $S, S_1, \ldots, S_m$ such that (a) for all $i = 1, \ldots, m$, \[ S \cdot (w - c) + S_i(w - c_i) \leq 0 \] for all worlds $w$ in $E_i$; and (b) for some $i = 1, \ldots, m$, \[ S \cdot (w - c) + S_i(w - c_i) < 0 \] for some world $w$ in $E_i$.
(ii) Suppose $c, c_1, \ldots, c_n$ satisfy Reflection.  Then there are no books $S, S_1, \ldots, S_m$ such that (a) for all $i = 1, \ldots, m$, \[S \cdot (w- c) + S_i\cdot (w-c_i) \leq 0\] for all worlds $w$ in $E_i$; and (b) there is $i = 1, \ldots, m$ such that \[S \cdot (w- c) + S_i\cdot (w-c_i) < 0 \] for some $w$ in $E_i$.

What does this say?  It says that, if you plan to update in some way other than conditioning on your evidence, and thereby violate Reflection, there is a book $S$ that you will accept at $t$ as well as, for each $E_i$, a book $S_i$ that you will accept at $t'$ if you learn $E_i$ such that, together, they will guarantee you a loss.  And this will not happen if you plan to update by conditioning.

How do we prove this?  Theorem 3(i) is the easier to prove.  Suppose $c, c_1, \ldots, c_n$ violate Reflection.  First, suppose that this is because $c_i(E_i) < 1$.  Then let $S = 0$ and $S_j = 0$ for all $j \neq i$.  And let $S_i$ be the book consisting only of a bet on $E_i$ with stake $-1$.  Then \[ S \cdot (w-c) + S_i(w-c_i) = (-1)(1 - c_i(E_i)) < 0\] for all worlds $w$ in $E_i$.  And \[ S \cdot (w-c) + S_i(w-c_i) = 0\] for all worlds $w$ in $E_j \neq E_i$.

Second, suppose that $c_i(E_i) = 1$ for all $i = 1, \ldots, m$.  But suppose $c$ is not inside the convex hull of the $c_i$s.  So $c, c_1, \ldots, c_n$ violate Reflection.  Then, adapting the proof of Theorem 1 by replacing the worlds $w$ with the planned posterior credences $c_i$, we get that there is a book $S$ such that \[ S \cdot (c_i - c) < 0\] for all $i = 1, \ldots, m$.  So if we let $S_i = -S$ for all $i = 1, \ldots, m$, we get \[ 0 > S \cdot (c_i - c) = S \cdot (w-c) + (-S)\cdot (w-c_i) = S \cdot (w-c) + S_i \cdot (w-c_i) \] for all worlds $w$.  This completes the proof of Theorem 3(i).

Now we turn to Theorem 3(ii).  Suppose $c, c_1, \ldots, c_n$ satisfy Reflection. Suppose, for a contradiction, that we have (a) for all $i = 1, \ldots, m$, \[S \cdot(w-c) + S_i \cdot(w-c_i) \leq 0 \] for all $w$ in $E_i$; and (b) for some $i = 1, \ldots, m$, \[S \cdot(w-c) + S_i \cdot(w-c_i) < 0 \] for some $w$ in $E_i$.  Our plan is to use this to construct $S'$ such that (a) for all $i = 1, \ldots, m$, \[S' \cdot(w-c) \leq 0\] for all $w$ in $E_i$; and (b) for some $i = 1, \ldots, m$, \[S' \cdot(w-c) < 0\] for some $w$ in $E_i$.  And we know that this is impossible from Theorem 1(ii).

We construct $S'$ as follows: First, suppose that $X_1, \ldots, X_k$ are the atoms of the algebra $\mathcal{F} = \{X_1, \ldots,  X_k, \ldots, X_n\}$.  Then notice that for each book of bets \[S = (S_1, \ldots, S_n)\] on the propositions $X_1, \ldots, X_n$, there is a book \[S^A = (S^A_1, \ldots, S^A_k, 0, \ldots, 0)\] on the atoms $X_1, \ldots, X_k$ of $\mathcal{F}$ such that $S^A$ is equivalent to $S$:  that is, the payout of $S$ is the same as the payout of $S^A$ at every world; and the price that a probabilistic agent should pay for $S^A$ is exactly the price she should pay for $S$.  Thus, if we have $S \cdot(w-c) + S_i \cdot(w-c_i) \leq 0$, then we have $S^A \cdot(w-c) + S^A_i \cdot(w-c_i) \leq 0$, and so on.  Thus, in what follows, we can assume without loss of generality that $S$ is a book of bets only on the atoms of $\mathcal{F}$.  Then we define $S'$ as follows, where for any atom $X_j$, we write $E_{i_j}$ for the cell of the partition in which $X_j$ lies:
\[
S'(X_j) := S(X_j) + S_{i_j}(X_j) - \sum_{X_k \in E_{i_j}} c(X_k | E_{i_j}) S_{i_j}(X_k)
\]
Then we can show that,
\[
S'\cdot(w - c) = S\cdot(w - c) + S_i\cdot (w - c_i)
\]
for all $E_i$ and $w \in E_i$.  Suppose $w$ is a world; suppose $X_j$ is the atom that is true at that world; suppose, as above, that $X_j$ lies in cell $E_{i_j}$.  Then we have
\begin{eqnarray*}
S'\cdot(w - c) & = & S(X_j) + S_{i_j}(X_j) - \sum_{X_k \in E_{i_j}} c(X_k | E_{i_j}) S_{i_j}(X_k) \\
& & \ \ \ - \sum_{X_l} c(X_l) [S(X_l) + S_{i_l}(X_l) - \sum_{X_k \in E_{i_l}} c(X_k | E_{i_l}) S_{i_l}(X_k)]\\
& = & S \cdot w + S_{i_j}\cdot w - S_{i_j} \cdot c_{i_j} - S \cdot c \\
 & & \ \ \ - \sum_{X_l} c(X_l) [S_{i_l}(X_l) - \sum_{X_k \in E_{i_l}} c(X_k | E_{i_l}) S_{i_l}(X_k)]\\
& = & S \cdot (w - c) + S_{i_j}\cdot (w - c_{i_j}) \\
 & & \ \ \ - \sum_{X_l} c(X_l) [S_{i_l}(X_l) - \sum_{X_k \in E_{i_l}} c(X_k | E_{i_l})S_{i_l}(X_k)]\\
& = & S \cdot (w - c) + S_{i_j}\cdot (w - c_{i_j}) < 0
\end{eqnarray*}
by assumption.  This completes the proof of Theorem 3(ii) and thus Theorem 3.

Thursday, 19 September 2013

Website: The Network for Philosophy of Mathematics in Europe

Paula Quinon (University of Lund, Sweden), Jan Heylen (KU Leuven, Belgium), and Fredrik Engström (University of Gothenburg, Sweden) have produced a wonderful new website listing people and activities in the philosophy of mathematics in Europe.

http://www.philmath-europe.org/

It includes lists of blogs on topics in philosophy of mathematics, scholarly associations, papers in the subject, researchers in the area, and jobs with philosophy of mathematics as an AOS.  It's quick and easy to register yourself as a researcher so that you appear on the list.  Thanks to Paula, Jan, and Fredrik for such a valuable service to the profession.

Wednesday, 18 September 2013

The Space of Languages

From time to time, I post things on the concept of language. What got me thinking about this originally was the modal status of T-sentences, and I gave a few talks on it over the last six or seven years, including a talk "Cognizing a Language" at the linguistics society in Edinburgh.

Since semantic concepts are language-dependent ("true-in-L", "refers-in-L", "implies-in-L", etc.), there's a quick argument that T-sentences are necessities. But this point is by no means restricted to T-sentences. The same holds when we consider any description of the syntactic, phonological, semantic, pragmatic properties of a language L. And, consequently, we need to distinguish:
(i) the syntactic, phonological, semantic, pragmatic properties of a language L
(ii) the cognizing relation that holds between an agent A and a language/idiolect L that A speaks, implements, realizes, etc.
because the modal status of the relevant facts is entirely different. (The orthodox view is that semantic facts are contingent. See, e.g., here.)

The basic argument was given by Field (1986) and Putnam (1985): for Putnam, in particular, applying modus tollens, it was some kind of reductio of Tarskian semantic theory that it yields the conclusion that semantic facts are necessities. But I apply modus ponens: they are necessities. What is contingent is the cognizing relation between agents and languages. Analogously, the properties of some Turing machine program are necessary, while it is contingent whether some physical machine "realizes" or "implements" that program.

A community $G$ of agents will, in general, all cognize different idiolects, $L_1, L_2, \dots$, even if they are very similar to each other. The point is that, strictly speaking, $L_i \neq L_j$ (for $i \neq j$). And a single agent may cognize multiple idiolects, or "micro-idiolects", which may be changing all the time. So, Humpty Dumptyism is true (... despite the protests of many philosophers!).

On this view, suppose we define $\Omega$ to be the space of all languages. So,
$\Omega$ = the collection of all $L$ such that $L$ is a language.
There may well be a Russell-style paradox, connected to largeness and self-reference, lurking here; maybe I'll mention it at some later point. So, $\Omega$ might have to be a somehow regulated space; e.g., the space of all set-sized, or maybe well-founded, languages.

$\Omega$ contains the idiolects spoken by each and every cognitive system, each human, old and young, each non-human creature, any non-terrestrial creature, or cognitive system there might be; as well as all the languages that, for feasibility reasons, cannot be spoken/cognized. $\Omega$ contains the idiolect you speak right now, and all other idiolects you may have spoken as your language state evolved to its present one. It contains all theoretically defined languages, finite and infinite, etc. It contains uninterpreted languages and it contains interpreted languages. It contains the Guitar Language, which is an odd language with no syntax at all.

[If so-called natural languages are languages (I think they aren't), then $\Omega$ contains all natural languages. But I think "natural languages" are idealized entities of some sort, as there is no individual that actually speaks or cognizes such a language. Strictly speaking, so-called natural languages, such as English, French, Hindi, etc., do not exist, in the sense of there being a community all speaking the same language. For example, what is the exact number of words in English? What is the exact pronunciation of "ouch"? Speakers exist and so do their idiolects, which may be changing in very complicated ways. But the concept of a natural language seems to be some kind of Hegelian myth, akin to "races".]

If what is said above is right (... I am plowing a very lonely furrow here), the sub-discipline within the philosophy of language that's now called "metasemantics" then has two main tasks. But these have a fundamentally different character modally and scientifically:
(i) What are the properties of the space $\Omega$ of languages? What are the individuation conditions for the elements $L \in \Omega$; what are the various relations amongst the $L \in \Omega$; etc.
(ii) How does the cognitive state of an agent evolve through $\Omega$? What is the nature of the cognizing relation, "$A$ cognizes $L$ (at time $t$)", which specifies the language state of a cognitive system $A$? How might it be constrained in terms of other cognitive states (memory, conceptual competence, perceptual input and action output, genetic factors, mental representation of strings and linguistic symbols, etc.) 
The first problem belongs to applied mathematics: and this seems to be well reflected by the actual practice of workers in this field. Languages $L \in \Omega$ are specified---usually by an explicit definition of their syntax and sometimes the meaning functions (referential, intensional and pragmatic)---and their properties are examined, usually by proving theorems. The Chomsky Hierarchy is an example, but there are literally countless examples: uninterpreted formal languages; simple propositional languages; predicate logic languages; languages with all kinds of extra gadgets and operators, modal, epistemic, temporal, etc., operators; typed-languages; higher-order ones; infinitary languages; highly finitary languages; languages with no syntax (cf., the Guitar Language); etc.; etc.

Let us say that those who work on the first problem are studying $\Omega$, the space of all languages. Modally speaking, the properties of languages $L \in \Omega$ established are essential. Relations amongst languages $L_1, L_2 \in \Omega$ hold of necessity. For example, true claims of the form
$L^{+}$ is an extension of $L$ such that there is a relation definable in $L^{+}$ but not in $L$.
There is no intension-preserving translation $t: L_1 \to L_2$.
The string $\sigma$ is true in $L$ if and only if snow is white.
will hold of necessity.

Some theoretical linguistics, formal semantics, computational linguistics, mathematical logic, etc., belongs to this area: they are studying $\Omega$. Their theorems are about $\Omega$. Their theorems hold of necessity. The semantic description of a language $L \in \Omega$ holds of necessity. For the semantic properties of $L$ are intrinsic to it. If $L^{\ast}$ has different semantic properties, then $L^{\ast} \neq L$.

On the other hand, the question:
Does $L \in \Omega$ have one, many or no agents that speak/cognize $L$?
is a contingent matter. Compare with, say, the questions:
Does the large natural number $10^{10^{10^{10}}}$ have a "physical token"?
Is the infinite cardinal $\aleph_0$ "physically realized"?
etc.
These are contingent matters, requiring physical theory and experiment to help answer them.

The second problem, in contrast, belongs to empirical science. But I think the problem(s) here are very difficult, much harder than those confronting the first problem. If we are honest, very little is known about:
  • the genetic basis of language cognition, 
  • how a cognitive language-using system evolves to anything like the mature cognitive state, 
  • what grounds or constitutes a cognitive system's cognizing $L$ rather than $L^{\ast}$, 
  • etc.
By analogy with physics, one would like to have some account of a "state-function'',
$L_A(t) \in \Omega$
which specifies how the language-cognizing state evolves, over time, through successive idiolects, and in connection with other states of the system. (Cf., in physics, the state of the system is an element of a state space, and the dynamical principles specify its evolution.)

The second problem uses (contingent) notions like:
$A$ "uses" string $\sigma$ when in a certain cognitive/affective state
$A$ and $B$ "communicate" with each other
$A$ "acquired" language by "interacting" with $B$
$A$ "copied" a word+meaning from $B$
$A$ introduced a new string $\sigma$.
$A$ "uses" string $n$ to refer to $x$.
etc.
It seems to me that no one properly understands any of this.

For example, how does one explain how agents $A$ and $B$ "communicate"? What is a "language community"? What could a "communal/social language" be? The cognitive states involved in a group of interacting speakers are associated with the idiolects actually spoken, much as in physics one is interested in the states that the system actually is in. What is the exact cognitive/affective state that an agent is in when "using" the strings "ouch" or "it's raining" or "the square root of $2$ is irrational"? What explains the introduction of new strings? How is a string "used" to refer to some object? To the object then, or the object now? Can anyone predict with some reasonable accuracy the evolving sequence of idiolects of a child? There are (combinatorially) countlessly many orbits through $\Omega$. Why one, and not another? What is the initial cognitive state? No one knows.

Winter School on Rationality - Groningen

(Potentially of interest to M-Phi readers and their students.)

On January 27th-28th 2014, the Faculty of Philosophy of the University of Groningen will host a short Winter School aimed at advanced undergraduate students and early-stage graduate students. The theme of the winter school is Rationality, and it will consist of 5 tutorials of 2 sessions each where the topic will be discussed from different viewpoints: theoretical rationality, practical rationality, and the history of the concept of rationality.

LECTURERS AND TUTORIALS

  • Catarina Dutilh Novaes: ‘Rationality and the psychology of reasoning’
  • Martin Lenz: ‘Rationalism in the history of philosophy’
  • Jan-Willem Romeijn: ‘Rationality and scientific method’
  • Bart Streumer: ‘Philosophical views on practical reasoning’
  • Peter Timmerman: 'Social contract theory and rationality' 

Moreover, Prof. Pauline Kleingeld may deliver a guest lecture (TBC).

As such, the program will showcase the high level of teaching and research of the three departments of the Faculty (theoretical philosophy; ethics, social and political philosophy; history of philosophy). The winter school aims in particular (but not exclusively) to attract potential talented students for our Research Masters’ program, who in this way will have the opportunity to become acquainted with the Faculty and the different lines of research we pursue.

SCHOLARSHIPS

The Faculty is offering up to three EUR 300 scholarships for the best students enrolling in the winter school, and who express serious interest in later applying for the Research Masters’ program. Moreover, participants who are then accepted in the Research Masters’ program for the year 2014/2015 will have their registration fee for the winter school reimbursed.
To apply for the scholarships, send a short CV (max 2 pages) and a letter (max 1 page) stating your interest in the Faculty of Philosophy in Groningen and the Research Masters’ program in particular, to winterschoolphilosophy 'at' rug.nl with 'Application for winter school scholarship' as subject. Deadline to apply for the scholarships: December 1st 2013. Preference will be given to members of underrepresented groups in philosophy (women, people of color, persons with disabilities etc.).

REGISTRATION

To register, send an email with your name, affiliation and status (undergraduate, graduate) to winterschoolphilosophy 'at' rug.nl with 'Registration for winter school' as subject, no later than December 15th 2013. As the number of spots is limited, you are encouraged to register early.

CONFERENCES

Another attractive feature of the winter school is the fact that two major international conferences will take place immediately after the school, which makes a trip to Groningen even more worthwhile for those coming from far away. These are:

Sixth Conference of the Dutch-Flemish Association of Analytic Philosophy (January 29-31)
Dutch Seminar in Early Modern Philosophy (January 29-30)

GENERAL INFORMATION
Dates: January 27th – 28th 2014
Scholarship application deadline: December 1st 2013
Registration deadline: December 15th 2013
Registration fee: EUR 50 (to be reimbursed for those later accepted in the ReMa program)

Further inquiries can be directed to Catarina Dutilh Novaes, c.dutilh.novaes 'at' rug.nl.

Tuesday, 17 September 2013

"Naturalism" in Metaphysics

Naturalistic metaphysics is fashionable amongst philosophers. A recent article,
Maclaurin, J., and Dyke, H. 2012.  "What is Analytic Metaphysics For?", Australasian Journal of Philosophy 90.
aims to articulate the concept of naturalistic metaphysics and to criticize its (alleged) opponent. The abstract begins:
Abstract
We divide analytic metaphysics into naturalistic and non-naturalistic metaphysics. The latter we define as any philosophical theory that makes some ontological claim (as opposed to conceptual claim), where that ontological claim has no observable consequences.
A response to this article appears in:
McLeod, M., and Parsons, J. 2013. “Maclaurin and Dyke on Analytic Metaphysics”, Australasian Journal of Philosophy 91.
whose abstract begins:
Abstract:
We argue that Maclaurin and Dyke's recent critique of non-naturalistic metaphysics suffers from difficulties analogous to those that caused trouble for earlier positivist critiques of metaphysics. Maclaurin and Dyke say that a theory is naturalistic iff it has observable consequences. Depending on the details of this criterion, either no theory counts as naturalistic or every theory does.
This seems right to me. The examples discussed by McLeod and Parsons come from basic philosophy of science: for example, auxiliary hypotheses and the difficulties involved in formulating some principle of verifiability. So, what is being promoted as "naturalistic metaphysics" looks like reheated positivism and faces exactly similar objections.

Here is a bit more to consider:
(1) $\nabla \cdot B = 0$
(2) $\exists X \forall y (y \in X \leftrightarrow \phi(y))$
Neither of these has "observable consequences". Since the magnetic field $B$ is not observable (not observable to the human eye), it follows that Maxwell's equation, $\nabla \cdot B = 0$, has no observable consequences (for this, we need to show its consistency). And one can show that any consequence of the (predicative version of) Comprehension Principle (2), in the restricted language (i.e., without the set/class quantifiers), is a logical truth. (The Comprehension Principle is what makes mathematics applicable. In addition to asserting the existence of objects of pure mathematics, e.g., $\mathbb{R}^3$ and $SU(3)$, we can also assert the existence of the objects of physics: (mixed) functions on spacetime (such as wavefunctions and fields), and sets of spacetime points, and sets of more mundane concreta, etc.).

Observing iron filings, the readings on a Hall probe, etc., doesn't count. One needs to state auxiliary claims about how the unobservable magnetic field $B$ is locally coupled to point charges and dipoles, along with a complicated network of idealized assumptions about how Hall probes work, etc.


In the linked video, you hear (1:02) the announcer say,
"As you see, the magnetic field forces the iron filings to line up along the lines of force ..."
This is an auxiliary assumption. This is how science works.

A number of basic, explanatory fundamental principles are given which do not refer to "observables" and have no observable consequences. To obtain observable consequences, one needs to add very complicated networks of auxiliary hypotheses.

Auxiliary hypotheses are deductively indispensable. It is usually safe to assume their truth, because the experimental setup normally---but after a lot of work, usually---ensures that the required idealizations are ok. But there are always cases where a failure of observation does not imply the law predicting it false. Rather, one of the auxiliaries is to blame. (This is called the Duhem-Quine Problem/Thesis.) This is not just logically obvious, it's also obvious to anyone who has worked with, e.g., an oscilloscope or pretty much any measuring device. For example:
Is the oscilloscope plugged in?
If you press a light switch and the light doesn't come on, the reasonable explanation is not that James Clark Maxwell has been refuted after all, but rather than some wire is not connected, or the bulb has blown, etc.

Similarly in the case of applicable mathematics and every physical principle of any interest (Maxwell's laws, Euler-Lagrange equations, laws of gravitation, principles of quantum theory, etc.). Although the basic principles of applicable mathematics have no observable consequences, it's still an interesting question to examine how the axioms of applicable mathematics interact with the mixed laws of physics to obtain measurable consequences. This is a non-trivial and not well-understood problem. It is more or less equivalent to Hilbert's 6th Problem:
6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.
Overall, why not simply accept that there is no opposition between analytic metaphysics---which I'm inclined to define, by ostension, as the writings of Frege, Moore and Russell + some similarity sphere---and science, or anything close to that? For example, look at the table of contents of Russell's Principles of Mathematics (1903).

Thursday, 12 September 2013

Second International Meeting of the Association for the Philosophy of Mathematical Practice - October 3 & 4

I'm passing on the announcement and programme of the Second International Meeting of the Association for the Philosophy of Mathematical Practice (APMP 2013) and the Fourteenth Midwest Philosophy of Mathematics Workshop (MWPMW 14) on behalf of Andy Arana.

---

I'm happy to distribute the schedule for the Second International Meeting of the Association for the Philosophy of Mathematical Practice (APMP 2013), to be held at the University of Illinois at Urbana-Champaign on October 3 and 4. You can get more information about the meeting at http://institucional.us.es/apmp/index_APMP2013.htm. A version of the schedule with abstracts for each talk can be read athttps://www.dropbox.com/s/j2qzw3txu4o0bit/APMP%202013%20schedule.pdf.

All participants are invited to join us for the conference dinner on Friday, October 4, as guests of the University of Illinois. Please contact me (aarana@illinois.edu) to let me know if you'd like to attend, so that I can get an accurate count of guests for dinner. Please let me know if you have any special dietary needs as well.

I hope you'll join us next month for APMP 2013, and also for the Fourteenth Midwest Philosophy of Mathematics Workshop (MWPMW 14) which will also be hosted at Illinois immediately after APMP 2013 on October 5 and 6. Information about MWPMW 14 can be read athttps://mdetlefsen.nd.edu/midwest-philmath-workshop/mwpmw-14/


Best wishes,

Andy
 
---

APMP 2013 schedule

All talks will take place on the second floor of the Levis Faculty Center (http://union.illinois.edu/levis/GettingHere.html)


Thursday, October 3

9:00–10:15am: Marcus Giaquinto, Department of Philosophy, University College London: ``Epistemic roles of visual experience in mathematical knowledge acquisition"

10:15–11:30am: Elaine Landry, Department of Philosophy, University of California, Davis: ``Plato was Not a Mathematical Platonist"

11:30–12:00pm: Break

12:00–12:35pm: Dirk Schlimm, Department of Philosophy, McGill University: ``The early development of Dedekind's notion of mapping"

12:00–12:35pm: Joachim Frans, Centre for Logic and Philosophy of Science, VUB Brussels and Erik Weber, Centre for Logic and Philosophy of Science, Ghent University: ``Mechanistic Explanation And Explanatory Proofs In Mathematics"

12:35–1:10pm: Emmylou Haffner, Department of History and Philosophy of Science, Université Paris Diderot - Paris 7: ``Arithmetization of mathematics, the Dedekindian way"

12:35–1:10pm: Andrew Aberdein, Department of Humanities and Communication, Florida Institute of Technology and Matthew Inglis, Mathematics Education Centre, Loughborough University: ``Explanation and Explication In Mathematical Practice"

1:10–2:30pm: Lunch break

2:30–3:05pm: John Baldwin, Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago: ``From Geometry to Algebra and Analysis"

2:30–3:05pm: Ashton Sperry-Taylor, Department of Philosophy, University of Missouri: ``Modeling The State Of Nature"

3:05–3:40pm: Sylvain Cabanacq, Department of History and Philosophy of Science, Université Paris Diderot - Paris 7: "Contrasting proofs: categorical tools, model-theoretical methods and the Manin-Mumford Conjecture"

3:05–3:40pm: Luca San Mauro, Center of Philosophy, Scuola Normale Superiore, Pisa: ``Algorithms, formalization and exactness: a philosophical study of the practical use of Church- Turing Thesis"

3:40–4:15pm: Break

4:15–5:30pm: Chris Pincock, Department of Philosophy, The Ohio State University: ``Felix Klein as a Prototype for the Philosophy of Mathematical Practice"

A meeting of APMP members follows.


Friday, October 4

9:00–10:15am: Colin McLarty, Department of Philosophy, Case Western Reserve University: ``Proofs in practice"

10:15–10:50am: Ken Manders, Department of Philosophy, University of Pittsburgh: ``Problems and Prospects for Philosophy of Mathematical Understanding"

10:15–10:50am: Ramzi Kebaili, Department of History and Philosophy of Science, Université Paris Diderot - Paris 7: ``Examples of `synthetic a priori' statements in mathematical practice"

10:50–11:20am: Break

11:20–11:55am: Susan Vineberg, Department of Philosophy, Wayne State University: ``Are There Objective Facts of Mathematical Depth?"

11:20–11:55am: Oran Magal, Department of Philosophy, McGill University: ``Teratological Investigations: What's in a monster?"

11:55–12:30pm: Philip Ehrlich, Department of Philosophy, Ohio University: ``A Re-examination of Zeno's Paradox of Extension"

11:55–12:30pm: Madeline Muntersbjorn, Department of Philosophy, University of Toledo: ``Cognitive Diversity and Mathematical Progress"

12:30–2:00pm: Lunch break

2:00–2:35pm: Michele Friend, Department of Philosophy, George Washington University: ``Using a Paraconsistent Formal Theory of Logic Metaphorically"

2:00–2:35pm: José Ferreirós, Department of Philosophy and Logic, University of Seville and Elías Fuentes Guillén, Department of Philosophy, Logic and Aesthetics, University of Salamanca: ``Bolzano's `Rein analytischer Beweis\dots' reconsidered"

2:35–3:10pm: Bernd Buldt, Department of Philosophy, Indiana University-Purdue University Fort Wayne: ``Mathematics—Some ``Practical" Insights"

2:35–3:10pm: Jacobo Asse Dayán, Program in Philosophy of Science, Universidad Nacional Autónoma de México: ``The Intentionality and Materiality of Mathematical Objects"

3:10–3:40pm: Break

3:40–4:15pm: Rochelle Gutiérrez, Department of Curriculum and Instruction, University of Illinois at Urbana-Champaign, ``What is Mathematics? The Roles of Ethnomathematics and Critical Mathematics in (Re)Defining Mathematics for the Field of Education"

3:40–4:15pm: Danielle Macbeth, Department of Philosophy, Haverford College: ``Rigor, Deduction, and Knowledge in the Practice of Mathematics"

4:15–5:30pm: Marco Panza, Institute of History and Philosophy of Science and Technology, Université Paris 1 Panthéon-Sorbonne: ``On the Epistemic Economy of Formal Definitions"

The conference dinner follows at 7pm at the ACES Library, Heritage Room.